<< Chapter < Page Chapter >> Page >

Note that if we don't have the Extremal Clause,  0.5, 1.5, 2.5, ... can be included in N, which is not what we want as the set of natural numbers.

Example 2. Definition of the Set of Nonnegative Even Numbers NE

The set NE is the set that satisfies the following three clauses:

Basis Clause: 0 ∈ NE

Inductive Clause: For any element x in NE, x + 2 is in NE.

Extremal Clause: Nothing is in NE unless it is obtained from the Basis and Inductive Clauses.

Example 3. Definition of the Set of Even Integers EI

The set EI is the set that satisfies the following three clauses:

Basis Clause: 0 ∈ EI

Inductive Clause: For any element x in EI, x + 2, and x - 2 are in EI.

Extremal Clause: Nothing is in EI unless it is obtained from the Basis and Inductive Clauses.

Example 4. Definition of the Set of Strings S over the alphabet {a,b} excepting empty string. This is the set of strings consisting of a's and b's such as abbab, bbabaa, etc.

The set S is the set that satisfies the following three clauses:

Basis Clause: a ∈ S, and b ∈ S.

Inductive Clause: For any element x in S, ax ∈ S, and bx ∈ S.

Here ax means the concatenation of a with x.

Extremal Clause: Nothing is in S unless it is obtained from the Basis and Inductive Clauses.

Tips for recursively defining a set:

For the "Basis Clause", try simplest elements in the set such as smallest numbers (0, or 1), simplest expressions, or shortest strings. Then see how other elements can be obtained from them, and generalize that generation process for the "Inductive Clause".

The set of propositions (propositional forms) can also be defined recursively.

Generalized set operations

As we saw earlier, union, intersection and Cartesian product of sets are associative. For example (A ∪ B) ∪ C = A ∪ (B ∪ C)

To denote either of these we often use A ∪ B ∪ C.

This can be generalized for the union of any finite number of sets as A1 ∪ A2 ∪.... ∪ An.

which we write as

       i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {}

This generalized union of sets can be rigorously defined as follows:

Definition ( i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} ):

Basis Clause: For n = 1, i = 1 n A i = A 1 size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } =A rSub { size 8{1} } } {} .

Inductive Clause:   i = 1 n + 1 A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n+1} } A rSub { size 8{i} } } {} = i = 1 n A i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} ∪ An+1

Similarly the generalized intersection i = 1 n A i size 12{ intersection rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} and generalized Cartesian product i = 1 n A i size 12{ times rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } } {} can be defined.

Based on these definitions, De Morgan's law on set union and intersection can also be generalized as follows:

Theorem (Generalized De Morgan)

i = 1 n A i ¯ = i = 1 n A i ¯ size 12{ {overline { union rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } }} = intersection rSub { size 8{i=1} } rSup { size 8{n} } {overline {A rSub { size 8{i} } }} } {} ,     and

i = 1 n A i ¯ = i = 1 n A i ¯ size 12{ {overline { intersection rSub { size 8{i=1} } rSup { size 8{n} } A rSub { size 8{i} } }} = union rSub { size 8{i=1} } rSup { size 8{n} } {overline {A rSub { size 8{i} } }} } {}

Proof: These can be proven by induction on n and are left as an exercise.

Recursive definition of function

Some functions can also be defined recursively.

Condition: The domain of the function you wish to define recursively must be a set defined recursively.

How to define function recursively: First the values of the function for the basis elements of the domain are specified. Then the value of the function at an element, say x, of the domain is defined using its value at the parent(s) of the element x.

A few examples are given below.

They are all on functions from integer to integer except the last one.

Example 5: The function f(n) = n! for natural numbers n can be defined recursively as follows:

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Discrete structures' conversation and receive update notifications?

Ask
Subramanian Divya
Start Quiz